I know that $D_{∞}^{(2)}$ is the set of all bijections: $f: \mathbb{Z}^{2}\rightarrow \mathbb{Z}^{2}$ such that $d(f(x),f(y))=d(x,y)$.
So the group multiplication is given by the composition of functions.
I also know that the distance between two points $x=(x_{1},x_{2}), y= (y_{1},y_{2})$ is given by: d(x,y)=$\sqrt{(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}}$
Now I want to know if $D_{∞}^{(2)}$ is virtually abelian. In other words if there is a subgroup H of $D_{∞}^{(2)}$ such that it is abelian ($\forall x,y\in H: xy=yx$) and has finite index ( $\dfrac{|D_{∞}^{(2)}|}{|H|}\neq ∞$).
I believe $D_{∞}^{(2)}$ is indeed virtually abelian because $\mathbb{Z}^{2}$ satisfies the criteria is this correct?
First of all note that the image of a straight line is still a straight line, otherwise the distance preserving would be violated (on non successive points). Secondly, note that the image of a grid is still a grid. Note that the only distance preserving bijections are translations, vertical and horizontal symmerties and rotations over mutiples of $90°$. Conclude that yoru group is the semidirect product of the translations $\mathbb{Z}^2$ over the dihedral group $D_8$ with eight elements.